Mister Exam

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Identical expressions
(x- one)/(x^(one / two)+ one)
(x minus 1) divide by (x to the power of (1 divide by 2) plus 1)
(x minus one) divide by (x to the power of (one divide by two) plus one)
(x-1)/(x(1/2)+1)
x-1/x1/2+1
x-1/x^1/2+1
(x-1) divide by (x^(1 divide by 2)+1)
(x-1)/(x^(1/2)+1)dx
Similar expressions
(x+1)/(x^(1/2)+1)
(x-1)/(x^(1/2)-1)

Solving integrals/ (x-1)/(x^(1/2)+1)

Integral of (x-1)/(x^(1/2)+1) dx

The solution

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  9             
  /             
 |              
 |    x - 1     
 |  --------- dx
 |    ___       
 |  \/ x  + 1   
 |              
/               
4

\int\limits_{4}^{9} \frac{x - 1}{\sqrt{x} + 1}\, dx

Integral((x - 1)/(sqrt(x) + 1), (x, 4, 9))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \sqrt{x}$ .
  
  Then let $du = \frac{dx}{2 \sqrt{x}}$ and substitute $du$ :
  
  $\int \left(2 u^{2} - 2 u\right)\, du$
  1. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 2 u^{2}\, du = 2 \int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $\frac{2 u^{3}}{3}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 2 u\right)\, du = - 2 \int u\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $- u^{2}$
    The result is: $\frac{2 u^{3}}{3} - u^{2}$
  Now substitute $u$ back in:
  
  $\frac{2 x^{\frac{3}{2}}}{3} - x$
Method #2
1. Rewrite the integrand:
  
  $\frac{x - 1}{\sqrt{x} + 1} = \frac{x}{\sqrt{x} + 1} - \frac{1}{\sqrt{x} + 1}$
2. Integrate term-by-term:
  1. Let $u = \sqrt{x}$ .
    
    Then let $du = \frac{dx}{2 \sqrt{x}}$ and substitute $2 du$ :
    
    $\int \frac{2 u^{3}}{u + 1}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{u^{3}}{u + 1}\, du = 2 \int \frac{u^{3}}{u + 1}\, du$
      
      Rewrite the integrand:
      
      $\frac{u^{3}}{u + 1} = u^{2} - u + 1 - \frac{1}{u + 1}$
      
      Integrate term-by-term:
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u\right)\, du = - \int u\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $- \frac{u^{2}}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{u + 1}\right)\, du = - \int \frac{1}{u + 1}\, du$
      
      Let $u = u + 1$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(u + 1 \right)}$
      
      So, the result is: $- \log{\left(u + 1 \right)}$
      
      The result is: $\frac{u^{3}}{3} - \frac{u^{2}}{2} + u - \log{\left(u + 1 \right)}$
      
      So, the result is: $\frac{2 u^{3}}{3} - u^{2} + 2 u - 2 \log{\left(u + 1 \right)}$
    Now substitute $u$ back in:
    
    $\frac{2 x^{\frac{3}{2}}}{3} + 2 \sqrt{x} - x - 2 \log{\left(\sqrt{x} + 1 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{\sqrt{x} + 1}\right)\, dx = - \int \frac{1}{\sqrt{x} + 1}\, dx$
    1. Let $u = \sqrt{x}$ .
      
      Then let $du = \frac{dx}{2 \sqrt{x}}$ and substitute $2 du$ :
      
      $\int \frac{2 u}{u + 1}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{u}{u + 1}\, du = 2 \int \frac{u}{u + 1}\, du$
      
      Rewrite the integrand:
      
      $\frac{u}{u + 1} = 1 - \frac{1}{u + 1}$
      
      Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{u + 1}\right)\, du = - \int \frac{1}{u + 1}\, du$
      
      Let $u = u + 1$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(u + 1 \right)}$
      
      So, the result is: $- \log{\left(u + 1 \right)}$
      
      The result is: $u - \log{\left(u + 1 \right)}$
      
      So, the result is: $2 u - 2 \log{\left(u + 1 \right)}$
      
      Now substitute $u$ back in:
      
      $2 \sqrt{x} - 2 \log{\left(\sqrt{x} + 1 \right)}$
    So, the result is: $- 2 \sqrt{x} + 2 \log{\left(\sqrt{x} + 1 \right)}$
  The result is: $\frac{2 x^{\frac{3}{2}}}{3} - x$
Add the constant of integration:

$\frac{2 x^{\frac{3}{2}}}{3} - x+ \mathrm{constant}$

The answer is:

\frac{2 x^{\frac{3}{2}}}{3} - x+ \mathrm{constant}

The answer (Indefinite) [src]

  /                             
 |                           3/2
 |   x - 1                2*x   
 | --------- dx = C - x + ------
 |   ___                    3   
 | \/ x  + 1                    
 |                              
/

\int \frac{x - 1}{\sqrt{x} + 1}\, dx = C + \frac{2 x^{\frac{3}{2}}}{3} - x

Simplify

The graph

Plot the graph f(x)

The answer [src]

23/3

\frac{23}{3}

23/3

\frac{23}{3}

23/3

Numerical answer [src]

7.66666666666667

7.66666666666667

The graph

Use the examples entering the upper and lower limits of integration.

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Identical expressions

Similar expressions

Integral of (x-1)/(x^(1/2)+1) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2